Higher Geometry and Trigonometry: Being the Third Part of a Series on Elementary and Higher Geometry, Trigonometry, and Mensuration, Containing Many Valuable Discoveries and Improvements in Mathematical Science, Especially in Relation to the Quatrature of the Circle, and Some Other Curves, as Well as the Cubature of Certain Curvilinear Solids; Designed as a Text-book for Collegiate and Academic Instruction, and as a Practical Compendium of Mensuration
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altitude axis base becomes calculated called centre circle circumference common consequently construction contain corresponding cosec cosine curve describe determine diameter difference distance divided draw drawn equal equation example expression formula Geom geometrical given gles greater half hence intersection Join known less logarithm major axis manner means measured meet multiplied negative objects observed obtained opposite parallel passes perpendicular plane pole position PROBLEM produced Prop proportional PROPOSITION pyramid quadrant quantities radius ratio remaining represent respectively right angles segment sides similar sine solid solution sphere spherical triangle square straight line suppose surface tables tangent THEOREM trian triangle triangle ABC trigonometrical vertex vertices whence whole
Page 68 - In the same way it may be proved that a : b : : sin. A : sin. B, and these two proportions may be written a : 6 : c : : sin. A : sin. B : sin. C. THEOREM III. t8. In any plane triangle, the sum of any two sides is to their difference as the tangent of half the sum of the opposite angles is to the tangent of half their difference. By Theorem II. we have a : b : : sin. A : sin. B.
Page 81 - N .-. by definition, x — x" is the logarithm of ^ ; that is to say, The logarithm of a fraction, or of the quotient of two numbers, is equal to the logarithm of the numerator minus the logarithm of the denominator. III. Raise both members of equation (1) to the nth power. N"=a".
Page 27 - The circumference of every circle is supposed to be divided into 360 equal parts...
Page 7 - The radius of a sphere is a straight line, drawn from the centre to any point of the...
Page 8 - A spherical triangle is a portion of the surface of a sphere, bounded by three arcs of great circles.
Page 14 - ... point of intersection, as a pole, and limited by the sides, produced if necessary. Let the angle BAC be formed by the two arcs AB, AC ; then...
Page 18 - For, if the arc AD be drawn from the vertex A to the middle point D of the base, the two triangles ABD, ACD will have all the sides of the one respectively equal to the corresponding sides of the other, namely, AD common, BD=DC, and AB= AC : hence by the last Proposition, their angles will be equal ; therefore B=C.